Problem 51 - Precalculus

Question

A point in rectangular coordinates is given. Convert the point to polar coordinates. $$(-3,4)$$

Step-by-Step Solution

Answer

The polar coordinates of the point (-3,4) are (5, 2.21 rad).

Step 1: Calculate r (the distance of the point from the origin)

The distance r from the origin to the point in rectangular coordinates (x,y) is given by the Pythagorean theorem: \( r = \sqrt{x^2 + y^2} \). For the point (-3,4), you can plug these values into the equation to get \( r = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \)

Step 2: Calculate θ (the angle)

Remember that angle θ can be calculated using the arctan function. However, you have to be careful here because just using the arctan function will only give you the correct angle in the first and fourth quadrants. Since the point (-3,4) is in the second quadrant, you'll have to add π to the result of the arctan function. Hence, θ = arctan(y/x) + π = arctan(4/-3) + π = -0.93 rad + π = 2.21 rad.

Step 3: Write the answer

Now that you have both r and θ, you can write down the point in polar coordinates as (r, θ).

Key Concepts

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are a way of representing points in a two-dimensional plane. You typically use two numbers to define a point,

• x: The horizontal component, measured along the x-axis
• y: The vertical component, measured along the y-axis

In essence, they allow you to describe the exact location of a point where two lines intersect on a graph. For instance, the point \((-3, 4)\) tells us that the point lies 3 units to the left of the origin (since it’s negative) and 4 units up from the origin. This coordinate system is highly intuitive and is often the first one we learn in mathematics because it mirrors our natural sense of up/down and left/right directions.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry. It relates the lengths of the sides of a right triangle. If a triangle is right-angled, then:\[a^2 + b^2 = c^2\]Where:

• a and b are the lengths of the legs of the triangle, perpendicular to each other.
• c is the length of the hypotenuse, the side opposite the right angle.

In the context of converting rectangular coordinates to polar coordinates, the Pythagorean theorem helps us calculate the radial distance \(r\) from the origin to the point. For example, with the point \((-3,4)\), the x-coordinate \(-3\) and the y-coordinate \(4\) are analogous to the triangle's legs. Thus, using Pythagorean theorem, we get:\[r = \sqrt{(-3)^2 + 4^2} = \sqrt{9 + 16} = 5\]This is how we determine how far the point is from the origin.

Arctan Function

The arctan function, or inverse tangent, is used to find an angle when given the ratio of the lengths of the opposite and adjacent sides of a right triangle. When converting from rectangular coordinates to polar coordinates, we use the arctan function to determine the angle \(\theta\). The formula looks like:\[\theta = \arctan\left(\frac{y}{x}\right)\]However, it's important to understand that arctan alone does not always provide the correct angle. This is because it typically returns values between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\), which cover only the first and fourth quadrants. So for points in the second quadrant, like \((-3, 4)\), you must add \(\pi\) to the angle given by the arctan function to obtain the correct result. This correction accounts for the initial direction and ensures your angle reflects the true position on the plane:\[\theta = \arctan\left(\frac{4}{-3}\right) + \pi = 2.21 \text{ rad}\]This adjustment ensures our calculations accurately represent the angle in terms of conventional polar coordinates.

Second Quadrant

The second quadrant in a Cartesian plane is where both x-coordinates are negative, and y-coordinates are positive. It is located in the top left section of the plane, spanning angles from \(\frac{\pi}{2}\) to \(\pi\). When converting rectangular coordinates to polar coordinates, identifying the quadrant in which the point lies is essential. This identification helps in adjusting the angle calculated with the arctan function to ensure it accurately represents the point's location. For example, the point \((-3, 4)\) is in the second quadrant because:

• The x-coordinate (-3) is negative, indicating leftward direction from the vertical axis.
• The y-coordinate (4) is positive, indicating upward direction from the horizontal axis.

Recognizing this allows us to correctly add \(\pi\) to the angle obtained from the arctan function, making sure our polar angle says "Hey, we're over here in the second quadrant!" This is crucial for converting and understanding polar coordinates properly.